from \(C^{\alpha }(\mathbb {T})\)  into itself (see [1]} for more details) added to the fact that \(g_{2}\in C^{\alpha }(\mathbb {T})\) , we deduce that \(g_{2}^{+}\in C^{\alpha }(\mathbb {T}).\)  Applying Bernstein Theorem of Fourier series gives that \(g_{2}^{+}\)  is the uniform limit of its Fourier series, namely
\(\Vert  S_{N}^{\prime }- \overline{w}g_{2}^{+}\Vert _{L^{\infty }(\mathbb {T})}\underset{N\rightarrow \infty }{\longrightarrow }0.\) 
